Quadratic overgroups for diamond Lie groups

Let G be a connected and simply connected solvable Lie group. The moment map for π in Gˆ, unitary dual of G, sends smooth vectors in the representation space of π to g⁎, dual space of g. The closure of the image of the moment map for π is called its moment set, denoted by Iπ. Generally, the moment set Iπ, π∈Gˆ does not characterize π, even for generic representations. However, we say that Gˆ is moment separable when the moment sets differ for any pair of distinct irreducible unitary representations. In the case of an exponential solvable Lie group G, D. Arnal and M. Selmi exhibited an accurate construction of an overgroup G+, containing G as a subgroup and an injective map Φ from Gˆ into G+ˆ in such a manner that Φ(Gˆ) is moment separable and IΦ(π) characterizes π, π∈Gˆ. In this work, we provide the existence of a quadratic overgroup for the diamond Lie group, which is the semi-direct product of Rn with (2n+1)-dimensional Heisenberg group for some n⩾1.